so that \(g=M^{-1}\,{\bar{g}}\)  and fix any \(M\) . Since \(|r|^{(l)}_{S(1,{\bar{g}})}\le C_lM^{-1}\) , from the Calderón-Vaillancour theorem we have \(\Vert {\rm op}(r)\Vert \le CM^{-1}\) . Therefore for large \(M\)  there exists the inverse \((1-{\rm op}({r}))^{-1}\)  which is given by \(1+\sum _{\ell =1}^{\infty }r^{\#\ell }\in S(1,{\bar{g}})\) . (see [1]}). Write \(k=\sum _{\ell =1}^{\infty }r^{\#\ell }\)  then it is easy to see from the proof (see, e.g. [2]}, [3]}) that there is \(M_0\)  such that
\(|k|^{(l)}_{S(1,{\bar{g}})}\le C_l\) 
